# Rationality of Reward Sharing in Multi-agent Reinforcement Learning

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## Abstract

In multi-agent reinforcement learning systems, it is important to share a reward among all agents. We focus on the
where

*Rationality Theorem of Profit Sharing*[5] and analyze how to share a reward among all profit sharing agents. When an agent gets a*direct reward R*(*R*> 0), an*indirect reward µR*(µ ≥ 0) is given to the other agents. We have derived the necessary and sufficient condition to preserve the rationality as follows$$
\mu < \frac{{M - 1}}
{{M^W \left( {1 - (\tfrac{1}
{M})^{W_0 } } \right)\left( {n - 1} \right)L}},
$$

*M*and*L*are the maximum number of conflicting all rules and rational rules in the same sensory input,*W*and*W*_{0}are the maximum episode length of a*direct*and an*indirect-reward*agents, and n is the number of agents. This theory is derived by avoiding the least desirable situation whose expected reward per an action is zero. Therefore, if we use this theorem, we can experience several efficient aspects of reward sharing. Through numerical examples, we confirm the effectiveness of this theorem.## Keywords

Sensory Input Rational Rule Rule Sequence Negative Reward Sharing Agent
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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